Question

Solution of Gamma function of 2n

Answer 1

The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by

Gamma(n)=(n-1)!,  

(1)

a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler Pi(n)=n! (Gauss 1812; Edwards 2001, p. 8).

It is analytic everywhere except at z=0, -1, -2, ..., and the residue at z=-k is

Res_(z=-k)Gamma(z)=((-1)^k)/(k!).  

(2)

There are no points z at which Gamma(z)=0.

The gamma function is implemented in the Wolfram Language as Gamma[z].

There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use Gamma^n(z) (i.e., using a trigonometric function-like convention), it is also common to write [Gamma(z)]^n.

The gamma function can be defined as a definite integral for R[z]>0 (Euler's integral form)

Gamma(z) = int_0^inftyt^(z-1)e^(-t)dt  

(3)

= 2int_0^inftye^(-t^2)t^(2z-1)dt,  

(4)

or

Gamma(z)=int_0^1[ln(1/t)]^(z-1)dt.  

(5)

The complete gamma function Gamma(x) can be generalized to the upper incomplete gamma function Gamma(a,x) and lower incomplete gamma function gamma(a,x).

GammaReImAbs

 Min  Max    

Re  

-5

5

Im  

-5

5

Plots of the real and imaginary parts of Gamma(z) in the complex plane are illustrated above.